Capturing the Macroscopic Behaviour of Molecular Dynamics with Membership Functions

Markov processes serve as foundational models in many scientific disciplines, such as molecular dynamics, and their simulation forms a common basis for analysis. While simulations produce useful trajectories, obtaining macroscopic information directly from microstate data presents significant challenges. This paper addresses this gap by introducing the concept of membership functions being the macrostates themselves. We derive equations for the holding times of these macrostates and demonstrate their consistency with the classical definition. Furthermore, we discuss the application of the ISOKANN method for learning these quantities from simulation data. In addition, we present a novel method for extracting transition paths from simulations based on the ISOKANN results and demonstrate its efficacy by applying it to simulations of the {\mu}-opioid receptor. With this approach we provide a new perspective on the analysis of macroscopic behaviour of Markov systems.